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The oscillation spectrum of a perturbed neutron star is intimately related tothe physical properties of the star, such as the equation of state. Observingpulsating neutron stars therefore allows one to place constraints on thesephysical properties. However, it is not obvious exactly how much can be learntfrom such measurements. If we observe for long enough, and precisely enough, isit possible to learn everything about the star? A classical result in thetheory of spectral geometry states that one cannot uniquely `hear the shape ofa drum'. More formally, it is known that an eigenfrequency spectrum may notuniquely correspond to a particular geometry; some `drums' may beindistinguishable from a normal-mode perspective. In contrast, we show that thedrum result does not extend to perturbations of simple neutron stars withingeneral relativity -- in the case of axial (toroidal) perturbations of static,perfect fluid stars, a quasi-normal mode spectrum uniquely corresponds to astellar profile. We show in this paper that it is not possible for two neutronstars, with distinct fluid profiles, to oscillate in an identical manner. Thisresult has the information-theoretic consequence that gravitational wavescompletely encode the properties of any given oscillating star: uniqueidentifications are possible in the limit of perfect measurement.

Astroseismology of neutron stars from gravitational waves in the limit of perfect measurement